Theorem chfacfscmulgsum 22353

Description: Breaking up a sum of values of the "characteristic factor function" scaled by a polynomial. (Contributed by AV, 9-Nov-2019.)

Hypotheses

Ref	Expression
chfacfisf.a	⊢ 𝐴 = (𝑁 Mat 𝑅)
chfacfisf.b	⊢ 𝐵 = (Base‘𝐴)
chfacfisf.p	⊢ 𝑃 = (Poly1‘𝑅)
chfacfisf.y	⊢ 𝑌 = (𝑁 Mat 𝑃)
chfacfisf.r	⊢ × = (.r‘𝑌)
chfacfisf.s	⊢ − = (-g‘𝑌)
chfacfisf.0	⊢ 0 = (0g‘𝑌)
chfacfisf.t	⊢ 𝑇 = (𝑁 matToPolyMat 𝑅)
chfacfisf.g	⊢ 𝐺 = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛))))))))
chfacfscmulcl.x	⊢ 𝑋 = (var1‘𝑅)
chfacfscmulcl.m	⊢ · = ( ·𝑠 ‘𝑌)
chfacfscmulcl.e	⊢ ↑ = (.g‘(mulGrp‘𝑃))
chfacfscmulgsum.p	⊢ + = (+g‘𝑌)

Assertion

Ref	Expression
chfacfscmulgsum	⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (𝑌 Σg (𝑖 ∈ ℕ0 ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)))) = ((𝑌 Σg (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ 𝑋) · ((𝑇‘(𝑏‘(𝑖 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖))))))) + ((((𝑠 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑠))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0))))))

Distinct variable groups:   𝐵,𝑛   𝑛,𝑀   𝑛,𝑁   𝑅,𝑛   𝑛,𝑌   𝑛,𝑏   𝑛,𝑠,𝐵   0 ,𝑛   𝐵,𝑖,𝑠   𝑖,𝐺   𝑖,𝑀   𝑖,𝑁   𝑅,𝑖   𝑖,𝑋   𝑖,𝑌   ↑ ,𝑖   · ,𝑏,𝑖   𝑇,𝑛   − ,𝑛   × ,𝑛   𝑖,𝑛

Allowed substitution hints:   𝐴(𝑖,𝑛,𝑠,𝑏)   𝐵(𝑏)   𝑃(𝑖,𝑛,𝑠,𝑏)   + (𝑖,𝑛,𝑠,𝑏)   𝑅(𝑠,𝑏)   𝑇(𝑖,𝑠,𝑏)   · (𝑛,𝑠)   × (𝑖,𝑠,𝑏)   ↑ (𝑛,𝑠,𝑏)   𝐺(𝑛,𝑠,𝑏)   𝑀(𝑠,𝑏)   − (𝑖,𝑠,𝑏)   𝑁(𝑠,𝑏)   𝑋(𝑛,𝑠,𝑏)   𝑌(𝑠,𝑏)   0 (𝑖,𝑠,𝑏)

Proof of Theorem chfacfscmulgsum

Step	Hyp	Ref	Expression
1		eqid 2732	. . 3 ⊢ (Base‘𝑌) = (Base‘𝑌)
2		chfacfisf.0	. . 3 ⊢ 0 = (0g‘𝑌)
3		chfacfscmulgsum.p	. . 3 ⊢ + = (+g‘𝑌)
4		crngring 20061	. . . . . . . 8 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
5	4	anim2i 617	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
6	5	3adant3 1132	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
7		chfacfisf.p	. . . . . . 7 ⊢ 𝑃 = (Poly1‘𝑅)
8		chfacfisf.y	. . . . . . 7 ⊢ 𝑌 = (𝑁 Mat 𝑃)
9	7, 8	pmatring 22185	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑌 ∈ Ring)
10	6, 9	syl 17	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑌 ∈ Ring)
11		ringcmn 20092	. . . . 5 ⊢ (𝑌 ∈ Ring → 𝑌 ∈ CMnd)
12	10, 11	syl 17	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑌 ∈ CMnd)
13	12	adantr 481	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → 𝑌 ∈ CMnd)
14		nn0ex 12474	. . . 4 ⊢ ℕ0 ∈ V
15	14	a1i 11	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → ℕ0 ∈ V)
16		simpll 765	. . . . 5 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑖 ∈ ℕ0) → (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵))
17		simplr 767	. . . . 5 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑖 ∈ ℕ0) → (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))))
18		simpr 485	. . . . 5 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑖 ∈ ℕ0) → 𝑖 ∈ ℕ0)
19	16, 17, 18	3jca 1128	. . . 4 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑖 ∈ ℕ0) → ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) ∧ 𝑖 ∈ ℕ0))
20		chfacfisf.a	. . . . 5 ⊢ 𝐴 = (𝑁 Mat 𝑅)
21		chfacfisf.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐴)
22		chfacfisf.r	. . . . 5 ⊢ × = (.r‘𝑌)
23		chfacfisf.s	. . . . 5 ⊢ − = (-g‘𝑌)
24		chfacfisf.t	. . . . 5 ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅)
25		chfacfisf.g	. . . . 5 ⊢ 𝐺 = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛))))))))
26		chfacfscmulcl.x	. . . . 5 ⊢ 𝑋 = (var1‘𝑅)
27		chfacfscmulcl.m	. . . . 5 ⊢ · = ( ·𝑠 ‘𝑌)
28		chfacfscmulcl.e	. . . . 5 ⊢ ↑ = (.g‘(mulGrp‘𝑃))
29	20, 21, 7, 8, 22, 23, 2, 24, 25, 26, 27, 28	chfacfscmulcl 22350	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) ∧ 𝑖 ∈ ℕ0) → ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)) ∈ (Base‘𝑌))
30	19, 29	syl 17	. . 3 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑖 ∈ ℕ0) → ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)) ∈ (Base‘𝑌))
31	20, 21, 7, 8, 22, 23, 2, 24, 25, 26, 27, 28	chfacfscmulfsupp 22352	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (𝑖 ∈ ℕ0 ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖))) finSupp 0 )
32		nn0disj 13613	. . . 4 ⊢ ((0...(𝑠 + 1)) ∩ (ℤ≥‘((𝑠 + 1) + 1))) = ∅
33	32	a1i 11	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → ((0...(𝑠 + 1)) ∩ (ℤ≥‘((𝑠 + 1) + 1))) = ∅)
34		nnnn0 12475	. . . . . 6 ⊢ (𝑠 ∈ ℕ → 𝑠 ∈ ℕ0)
35		peano2nn0 12508	. . . . . 6 ⊢ (𝑠 ∈ ℕ0 → (𝑠 + 1) ∈ ℕ0)
36	34, 35	syl 17	. . . . 5 ⊢ (𝑠 ∈ ℕ → (𝑠 + 1) ∈ ℕ0)
37		nn0split 13612	. . . . 5 ⊢ ((𝑠 + 1) ∈ ℕ0 → ℕ0 = ((0...(𝑠 + 1)) ∪ (ℤ≥‘((𝑠 + 1) + 1))))
38	36, 37	syl 17	. . . 4 ⊢ (𝑠 ∈ ℕ → ℕ0 = ((0...(𝑠 + 1)) ∪ (ℤ≥‘((𝑠 + 1) + 1))))
39	38	ad2antrl 726	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → ℕ0 = ((0...(𝑠 + 1)) ∪ (ℤ≥‘((𝑠 + 1) + 1))))
40	1, 2, 3, 13, 15, 30, 31, 33, 39	gsumsplit2 19791	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (𝑌 Σg (𝑖 ∈ ℕ0 ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)))) = ((𝑌 Σg (𝑖 ∈ (0...(𝑠 + 1)) ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)))) + (𝑌 Σg (𝑖 ∈ (ℤ≥‘((𝑠 + 1) + 1)) ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖))))))
41		simpll 765	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑖 ∈ (ℤ≥‘((𝑠 + 1) + 1))) → (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵))
42		simplr 767	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑖 ∈ (ℤ≥‘((𝑠 + 1) + 1))) → (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))))
43		nncn 12216	. . . . . . . . . . . . 13 ⊢ (𝑠 ∈ ℕ → 𝑠 ∈ ℂ)
44		add1p1 12459	. . . . . . . . . . . . 13 ⊢ (𝑠 ∈ ℂ → ((𝑠 + 1) + 1) = (𝑠 + 2))
45	43, 44	syl 17	. . . . . . . . . . . 12 ⊢ (𝑠 ∈ ℕ → ((𝑠 + 1) + 1) = (𝑠 + 2))
46	45	ad2antrl 726	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → ((𝑠 + 1) + 1) = (𝑠 + 2))
47	46	fveq2d 6892	. . . . . . . . . 10 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (ℤ≥‘((𝑠 + 1) + 1)) = (ℤ≥‘(𝑠 + 2)))
48	47	eleq2d 2819	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (𝑖 ∈ (ℤ≥‘((𝑠 + 1) + 1)) ↔ 𝑖 ∈ (ℤ≥‘(𝑠 + 2))))
49	48	biimpa 477	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑖 ∈ (ℤ≥‘((𝑠 + 1) + 1))) → 𝑖 ∈ (ℤ≥‘(𝑠 + 2)))
50	20, 21, 7, 8, 22, 23, 2, 24, 25, 26, 27, 28	chfacfscmul0 22351	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) ∧ 𝑖 ∈ (ℤ≥‘(𝑠 + 2))) → ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)) = 0 )
51	41, 42, 49, 50	syl3anc 1371	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑖 ∈ (ℤ≥‘((𝑠 + 1) + 1))) → ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)) = 0 )
52	51	mpteq2dva 5247	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (𝑖 ∈ (ℤ≥‘((𝑠 + 1) + 1)) ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖))) = (𝑖 ∈ (ℤ≥‘((𝑠 + 1) + 1)) ↦ 0 ))
53	52	oveq2d 7421	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (𝑌 Σg (𝑖 ∈ (ℤ≥‘((𝑠 + 1) + 1)) ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)))) = (𝑌 Σg (𝑖 ∈ (ℤ≥‘((𝑠 + 1) + 1)) ↦ 0 )))
54	4, 9	sylan2 593	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑌 ∈ Ring)
55		ringmnd 20059	. . . . . . . . . 10 ⊢ (𝑌 ∈ Ring → 𝑌 ∈ Mnd)
56	54, 55	syl 17	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑌 ∈ Mnd)
57	56	3adant3 1132	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑌 ∈ Mnd)
58		fvex 6901	. . . . . . . 8 ⊢ (ℤ≥‘((𝑠 + 1) + 1)) ∈ V
59	57, 58	jctir 521	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝑌 ∈ Mnd ∧ (ℤ≥‘((𝑠 + 1) + 1)) ∈ V))
60	59	adantr 481	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (𝑌 ∈ Mnd ∧ (ℤ≥‘((𝑠 + 1) + 1)) ∈ V))
61	2	gsumz 18713	. . . . . 6 ⊢ ((𝑌 ∈ Mnd ∧ (ℤ≥‘((𝑠 + 1) + 1)) ∈ V) → (𝑌 Σg (𝑖 ∈ (ℤ≥‘((𝑠 + 1) + 1)) ↦ 0 )) = 0 )
62	60, 61	syl 17	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (𝑌 Σg (𝑖 ∈ (ℤ≥‘((𝑠 + 1) + 1)) ↦ 0 )) = 0 )
63	53, 62	eqtrd 2772	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (𝑌 Σg (𝑖 ∈ (ℤ≥‘((𝑠 + 1) + 1)) ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)))) = 0 )
64	63	oveq2d 7421	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → ((𝑌 Σg (𝑖 ∈ (0...(𝑠 + 1)) ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)))) + (𝑌 Σg (𝑖 ∈ (ℤ≥‘((𝑠 + 1) + 1)) ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖))))) = ((𝑌 Σg (𝑖 ∈ (0...(𝑠 + 1)) ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)))) + 0 ))
65		fzfid 13934	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (0...(𝑠 + 1)) ∈ Fin)
66		elfznn0 13590	. . . . . . . 8 ⊢ (𝑖 ∈ (0...(𝑠 + 1)) → 𝑖 ∈ ℕ0)
67	66, 19	sylan2 593	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑖 ∈ (0...(𝑠 + 1))) → ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) ∧ 𝑖 ∈ ℕ0))
68	67, 29	syl 17	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑖 ∈ (0...(𝑠 + 1))) → ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)) ∈ (Base‘𝑌))
69	68	ralrimiva 3146	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → ∀𝑖 ∈ (0...(𝑠 + 1))((𝑖 ↑ 𝑋) · (𝐺‘𝑖)) ∈ (Base‘𝑌))
70	1, 13, 65, 69	gsummptcl 19829	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (𝑌 Σg (𝑖 ∈ (0...(𝑠 + 1)) ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)))) ∈ (Base‘𝑌))
71	1, 3, 2	mndrid 18642	. . . 4 ⊢ ((𝑌 ∈ Mnd ∧ (𝑌 Σg (𝑖 ∈ (0...(𝑠 + 1)) ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)))) ∈ (Base‘𝑌)) → ((𝑌 Σg (𝑖 ∈ (0...(𝑠 + 1)) ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)))) + 0 ) = (𝑌 Σg (𝑖 ∈ (0...(𝑠 + 1)) ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)))))
72	57, 70, 71	syl2an2r 683	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → ((𝑌 Σg (𝑖 ∈ (0...(𝑠 + 1)) ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)))) + 0 ) = (𝑌 Σg (𝑖 ∈ (0...(𝑠 + 1)) ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)))))
73	64, 72	eqtrd 2772	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → ((𝑌 Σg (𝑖 ∈ (0...(𝑠 + 1)) ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)))) + (𝑌 Σg (𝑖 ∈ (ℤ≥‘((𝑠 + 1) + 1)) ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖))))) = (𝑌 Σg (𝑖 ∈ (0...(𝑠 + 1)) ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)))))
74	34	ad2antrl 726	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → 𝑠 ∈ ℕ0)
75	1, 3, 13, 74, 68	gsummptfzsplit 19794	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (𝑌 Σg (𝑖 ∈ (0...(𝑠 + 1)) ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)))) = ((𝑌 Σg (𝑖 ∈ (0...𝑠) ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)))) + (𝑌 Σg (𝑖 ∈ {(𝑠 + 1)} ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖))))))
76		elfznn0 13590	. . . . . . 7 ⊢ (𝑖 ∈ (0...𝑠) → 𝑖 ∈ ℕ0)
77	76, 30	sylan2 593	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)) ∈ (Base‘𝑌))
78	1, 3, 13, 74, 77	gsummptfzsplitl 19795	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (𝑌 Σg (𝑖 ∈ (0...𝑠) ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)))) = ((𝑌 Σg (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)))) + (𝑌 Σg (𝑖 ∈ {0} ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖))))))
79	57	adantr 481	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → 𝑌 ∈ Mnd)
80		0nn0 12483	. . . . . . . 8 ⊢ 0 ∈ ℕ0
81	80	a1i 11	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → 0 ∈ ℕ0)
82	20, 21, 7, 8, 22, 23, 2, 24, 25, 26, 27, 28	chfacfscmulcl 22350	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) ∧ 0 ∈ ℕ0) → ((0 ↑ 𝑋) · (𝐺‘0)) ∈ (Base‘𝑌))
83	81, 82	mpd3an3 1462	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → ((0 ↑ 𝑋) · (𝐺‘0)) ∈ (Base‘𝑌))
84		oveq1 7412	. . . . . . . . 9 ⊢ (𝑖 = 0 → (𝑖 ↑ 𝑋) = (0 ↑ 𝑋))
85		fveq2 6888	. . . . . . . . 9 ⊢ (𝑖 = 0 → (𝐺‘𝑖) = (𝐺‘0))
86	84, 85	oveq12d 7423	. . . . . . . 8 ⊢ (𝑖 = 0 → ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)) = ((0 ↑ 𝑋) · (𝐺‘0)))
87	1, 86	gsumsn 19816	. . . . . . 7 ⊢ ((𝑌 ∈ Mnd ∧ 0 ∈ ℕ0 ∧ ((0 ↑ 𝑋) · (𝐺‘0)) ∈ (Base‘𝑌)) → (𝑌 Σg (𝑖 ∈ {0} ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)))) = ((0 ↑ 𝑋) · (𝐺‘0)))
88	79, 81, 83, 87	syl3anc 1371	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (𝑌 Σg (𝑖 ∈ {0} ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)))) = ((0 ↑ 𝑋) · (𝐺‘0)))
89	88	oveq2d 7421	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → ((𝑌 Σg (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)))) + (𝑌 Σg (𝑖 ∈ {0} ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖))))) = ((𝑌 Σg (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)))) + ((0 ↑ 𝑋) · (𝐺‘0))))
90	78, 89	eqtrd 2772	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (𝑌 Σg (𝑖 ∈ (0...𝑠) ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)))) = ((𝑌 Σg (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)))) + ((0 ↑ 𝑋) · (𝐺‘0))))
91		ovexd 7440	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (𝑠 + 1) ∈ V)
92		1nn0 12484	. . . . . . . 8 ⊢ 1 ∈ ℕ0
93	92	a1i 11	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → 1 ∈ ℕ0)
94	74, 93	nn0addcld 12532	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (𝑠 + 1) ∈ ℕ0)
95	20, 21, 7, 8, 22, 23, 2, 24, 25, 26, 27, 28	chfacfscmulcl 22350	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) ∧ (𝑠 + 1) ∈ ℕ0) → (((𝑠 + 1) ↑ 𝑋) · (𝐺‘(𝑠 + 1))) ∈ (Base‘𝑌))
96	94, 95	mpd3an3 1462	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (((𝑠 + 1) ↑ 𝑋) · (𝐺‘(𝑠 + 1))) ∈ (Base‘𝑌))
97		oveq1 7412	. . . . . . 7 ⊢ (𝑖 = (𝑠 + 1) → (𝑖 ↑ 𝑋) = ((𝑠 + 1) ↑ 𝑋))
98		fveq2 6888	. . . . . . 7 ⊢ (𝑖 = (𝑠 + 1) → (𝐺‘𝑖) = (𝐺‘(𝑠 + 1)))
99	97, 98	oveq12d 7423	. . . . . 6 ⊢ (𝑖 = (𝑠 + 1) → ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)) = (((𝑠 + 1) ↑ 𝑋) · (𝐺‘(𝑠 + 1))))
100	1, 99	gsumsn 19816	. . . . 5 ⊢ ((𝑌 ∈ Mnd ∧ (𝑠 + 1) ∈ V ∧ (((𝑠 + 1) ↑ 𝑋) · (𝐺‘(𝑠 + 1))) ∈ (Base‘𝑌)) → (𝑌 Σg (𝑖 ∈ {(𝑠 + 1)} ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)))) = (((𝑠 + 1) ↑ 𝑋) · (𝐺‘(𝑠 + 1))))
101	79, 91, 96, 100	syl3anc 1371	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (𝑌 Σg (𝑖 ∈ {(𝑠 + 1)} ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)))) = (((𝑠 + 1) ↑ 𝑋) · (𝐺‘(𝑠 + 1))))
102	90, 101	oveq12d 7423	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → ((𝑌 Σg (𝑖 ∈ (0...𝑠) ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)))) + (𝑌 Σg (𝑖 ∈ {(𝑠 + 1)} ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖))))) = (((𝑌 Σg (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)))) + ((0 ↑ 𝑋) · (𝐺‘0))) + (((𝑠 + 1) ↑ 𝑋) · (𝐺‘(𝑠 + 1)))))
103		fzfid 13934	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (1...𝑠) ∈ Fin)
104		simpll 765	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵))
105		simplr 767	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))))
106		elfznn 13526	. . . . . . . . . 10 ⊢ (𝑖 ∈ (1...𝑠) → 𝑖 ∈ ℕ)
107	106	nnnn0d 12528	. . . . . . . . 9 ⊢ (𝑖 ∈ (1...𝑠) → 𝑖 ∈ ℕ0)
108	107	adantl 482	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → 𝑖 ∈ ℕ0)
109	104, 105, 108, 29	syl3anc 1371	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)) ∈ (Base‘𝑌))
110	109	ralrimiva 3146	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → ∀𝑖 ∈ (1...𝑠)((𝑖 ↑ 𝑋) · (𝐺‘𝑖)) ∈ (Base‘𝑌))
111	1, 13, 103, 110	gsummptcl 19829	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (𝑌 Σg (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)))) ∈ (Base‘𝑌))
112	1, 3	mndass 18630	. . . . 5 ⊢ ((𝑌 ∈ Mnd ∧ ((𝑌 Σg (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)))) ∈ (Base‘𝑌) ∧ ((0 ↑ 𝑋) · (𝐺‘0)) ∈ (Base‘𝑌) ∧ (((𝑠 + 1) ↑ 𝑋) · (𝐺‘(𝑠 + 1))) ∈ (Base‘𝑌))) → (((𝑌 Σg (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)))) + ((0 ↑ 𝑋) · (𝐺‘0))) + (((𝑠 + 1) ↑ 𝑋) · (𝐺‘(𝑠 + 1)))) = ((𝑌 Σg (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)))) + (((0 ↑ 𝑋) · (𝐺‘0)) + (((𝑠 + 1) ↑ 𝑋) · (𝐺‘(𝑠 + 1))))))
113	79, 111, 83, 96, 112	syl13anc 1372	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (((𝑌 Σg (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)))) + ((0 ↑ 𝑋) · (𝐺‘0))) + (((𝑠 + 1) ↑ 𝑋) · (𝐺‘(𝑠 + 1)))) = ((𝑌 Σg (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)))) + (((0 ↑ 𝑋) · (𝐺‘0)) + (((𝑠 + 1) ↑ 𝑋) · (𝐺‘(𝑠 + 1))))))
114	106	nnne0d 12258	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∈ (1...𝑠) → 𝑖 ≠ 0)
115	114	ad2antlr 725	. . . . . . . . . . . . 13 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) → 𝑖 ≠ 0)
116		neeq1 3003	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑖 → (𝑛 ≠ 0 ↔ 𝑖 ≠ 0))
117	116	adantl 482	. . . . . . . . . . . . 13 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) → (𝑛 ≠ 0 ↔ 𝑖 ≠ 0))
118	115, 117	mpbird 256	. . . . . . . . . . . 12 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) → 𝑛 ≠ 0)
119		eqneqall 2951	. . . . . . . . . . . 12 ⊢ (𝑛 = 0 → (𝑛 ≠ 0 → 0 = (𝑇‘(𝑏‘(𝑖 − 1)))))
120	118, 119	mpan9 507	. . . . . . . . . . 11 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ 𝑛 = 0) → 0 = (𝑇‘(𝑏‘(𝑖 − 1))))
121		simplr 767	. . . . . . . . . . . . . . 15 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ 𝑛 = 0) → 𝑛 = 𝑖)
122		eqeq1 2736	. . . . . . . . . . . . . . . . 17 ⊢ (0 = 𝑛 → (0 = 𝑖 ↔ 𝑛 = 𝑖))
123	122	eqcoms 2740	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 0 → (0 = 𝑖 ↔ 𝑛 = 𝑖))
124	123	adantl 482	. . . . . . . . . . . . . . 15 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ 𝑛 = 0) → (0 = 𝑖 ↔ 𝑛 = 𝑖))
125	121, 124	mpbird 256	. . . . . . . . . . . . . 14 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ 𝑛 = 0) → 0 = 𝑖)
126	125	fveq2d 6892	. . . . . . . . . . . . 13 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ 𝑛 = 0) → (𝑏‘0) = (𝑏‘𝑖))
127	126	fveq2d 6892	. . . . . . . . . . . 12 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ 𝑛 = 0) → (𝑇‘(𝑏‘0)) = (𝑇‘(𝑏‘𝑖)))
128	127	oveq2d 7421	. . . . . . . . . . 11 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ 𝑛 = 0) → ((𝑇‘𝑀) × (𝑇‘(𝑏‘0))) = ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖))))
129	120, 128	oveq12d 7423	. . . . . . . . . 10 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ 𝑛 = 0) → ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))) = ((𝑇‘(𝑏‘(𝑖 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖)))))
130		elfz2 13487	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 ∈ (1...𝑠) ↔ ((1 ∈ ℤ ∧ 𝑠 ∈ ℤ ∧ 𝑖 ∈ ℤ) ∧ (1 ≤ 𝑖 ∧ 𝑖 ≤ 𝑠)))
131		zleltp1 12609	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑖 ∈ ℤ ∧ 𝑠 ∈ ℤ) → (𝑖 ≤ 𝑠 ↔ 𝑖 < (𝑠 + 1)))
132	131	ancoms 459	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑠 ∈ ℤ ∧ 𝑖 ∈ ℤ) → (𝑖 ≤ 𝑠 ↔ 𝑖 < (𝑠 + 1)))
133	132	3adant1 1130	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((1 ∈ ℤ ∧ 𝑠 ∈ ℤ ∧ 𝑖 ∈ ℤ) → (𝑖 ≤ 𝑠 ↔ 𝑖 < (𝑠 + 1)))
134	133	biimpcd 248	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑖 ≤ 𝑠 → ((1 ∈ ℤ ∧ 𝑠 ∈ ℤ ∧ 𝑖 ∈ ℤ) → 𝑖 < (𝑠 + 1)))
135	134	adantl 482	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((1 ≤ 𝑖 ∧ 𝑖 ≤ 𝑠) → ((1 ∈ ℤ ∧ 𝑠 ∈ ℤ ∧ 𝑖 ∈ ℤ) → 𝑖 < (𝑠 + 1)))
136	135	impcom 408	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((1 ∈ ℤ ∧ 𝑠 ∈ ℤ ∧ 𝑖 ∈ ℤ) ∧ (1 ≤ 𝑖 ∧ 𝑖 ≤ 𝑠)) → 𝑖 < (𝑠 + 1))
137	136	orcd 871	. . . . . . . . . . . . . . . . . . 19 ⊢ (((1 ∈ ℤ ∧ 𝑠 ∈ ℤ ∧ 𝑖 ∈ ℤ) ∧ (1 ≤ 𝑖 ∧ 𝑖 ≤ 𝑠)) → (𝑖 < (𝑠 + 1) ∨ (𝑠 + 1) < 𝑖))
138		zre 12558	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑠 ∈ ℤ → 𝑠 ∈ ℝ)
139		1red 11211	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑠 ∈ ℤ → 1 ∈ ℝ)
140	138, 139	readdcld 11239	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑠 ∈ ℤ → (𝑠 + 1) ∈ ℝ)
141		zre 12558	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑖 ∈ ℤ → 𝑖 ∈ ℝ)
142	140, 141	anim12ci 614	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑠 ∈ ℤ ∧ 𝑖 ∈ ℤ) → (𝑖 ∈ ℝ ∧ (𝑠 + 1) ∈ ℝ))
143	142	3adant1 1130	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((1 ∈ ℤ ∧ 𝑠 ∈ ℤ ∧ 𝑖 ∈ ℤ) → (𝑖 ∈ ℝ ∧ (𝑠 + 1) ∈ ℝ))
144		lttri2 11292	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑖 ∈ ℝ ∧ (𝑠 + 1) ∈ ℝ) → (𝑖 ≠ (𝑠 + 1) ↔ (𝑖 < (𝑠 + 1) ∨ (𝑠 + 1) < 𝑖)))
145	143, 144	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((1 ∈ ℤ ∧ 𝑠 ∈ ℤ ∧ 𝑖 ∈ ℤ) → (𝑖 ≠ (𝑠 + 1) ↔ (𝑖 < (𝑠 + 1) ∨ (𝑠 + 1) < 𝑖)))
146	145	adantr 481	. . . . . . . . . . . . . . . . . . 19 ⊢ (((1 ∈ ℤ ∧ 𝑠 ∈ ℤ ∧ 𝑖 ∈ ℤ) ∧ (1 ≤ 𝑖 ∧ 𝑖 ≤ 𝑠)) → (𝑖 ≠ (𝑠 + 1) ↔ (𝑖 < (𝑠 + 1) ∨ (𝑠 + 1) < 𝑖)))
147	137, 146	mpbird 256	. . . . . . . . . . . . . . . . . 18 ⊢ (((1 ∈ ℤ ∧ 𝑠 ∈ ℤ ∧ 𝑖 ∈ ℤ) ∧ (1 ≤ 𝑖 ∧ 𝑖 ≤ 𝑠)) → 𝑖 ≠ (𝑠 + 1))
148	130, 147	sylbi 216	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 ∈ (1...𝑠) → 𝑖 ≠ (𝑠 + 1))
149	148	ad2antlr 725	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) → 𝑖 ≠ (𝑠 + 1))
150		neeq1 3003	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝑖 → (𝑛 ≠ (𝑠 + 1) ↔ 𝑖 ≠ (𝑠 + 1)))
151	150	adantl 482	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) → (𝑛 ≠ (𝑠 + 1) ↔ 𝑖 ≠ (𝑠 + 1)))
152	149, 151	mpbird 256	. . . . . . . . . . . . . . 15 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) → 𝑛 ≠ (𝑠 + 1))
153	152	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ ¬ 𝑛 = 0) → 𝑛 ≠ (𝑠 + 1))
154	153	neneqd 2945	. . . . . . . . . . . . 13 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ ¬ 𝑛 = 0) → ¬ 𝑛 = (𝑠 + 1))
155	154	pm2.21d 121	. . . . . . . . . . . 12 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ ¬ 𝑛 = 0) → (𝑛 = (𝑠 + 1) → (𝑇‘(𝑏‘𝑠)) = ((𝑇‘(𝑏‘(𝑖 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖))))))
156	155	imp 407	. . . . . . . . . . 11 ⊢ (((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ ¬ 𝑛 = 0) ∧ 𝑛 = (𝑠 + 1)) → (𝑇‘(𝑏‘𝑠)) = ((𝑇‘(𝑏‘(𝑖 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖)))))
157	106	nnred 12223	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 ∈ (1...𝑠) → 𝑖 ∈ ℝ)
158		eleq1w 2816	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 = 𝑖 → (𝑛 ∈ ℝ ↔ 𝑖 ∈ ℝ))
159	157, 158	syl5ibrcom 246	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 ∈ (1...𝑠) → (𝑛 = 𝑖 → 𝑛 ∈ ℝ))
160	159	adantl 482	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → (𝑛 = 𝑖 → 𝑛 ∈ ℝ))
161	160	imp 407	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) → 𝑛 ∈ ℝ)
162	74	nn0red 12529	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → 𝑠 ∈ ℝ)
163	162	ad2antrr 724	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) → 𝑠 ∈ ℝ)
164		1red 11211	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) → 1 ∈ ℝ)
165	163, 164	readdcld 11239	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) → (𝑠 + 1) ∈ ℝ)
166	130, 136	sylbi 216	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 ∈ (1...𝑠) → 𝑖 < (𝑠 + 1))
167	166	ad2antlr 725	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) → 𝑖 < (𝑠 + 1))
168		breq1 5150	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = 𝑖 → (𝑛 < (𝑠 + 1) ↔ 𝑖 < (𝑠 + 1)))
169	168	adantl 482	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) → (𝑛 < (𝑠 + 1) ↔ 𝑖 < (𝑠 + 1)))
170	167, 169	mpbird 256	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) → 𝑛 < (𝑠 + 1))
171	161, 165, 170	ltnsymd 11359	. . . . . . . . . . . . . . 15 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) → ¬ (𝑠 + 1) < 𝑛)
172	171	pm2.21d 121	. . . . . . . . . . . . . 14 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) → ((𝑠 + 1) < 𝑛 → 0 = ((𝑇‘(𝑏‘(𝑖 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖))))))
173	172	ad2antrr 724	. . . . . . . . . . . . 13 ⊢ (((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ ¬ 𝑛 = 0) ∧ ¬ 𝑛 = (𝑠 + 1)) → ((𝑠 + 1) < 𝑛 → 0 = ((𝑇‘(𝑏‘(𝑖 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖))))))
174	173	imp 407	. . . . . . . . . . . 12 ⊢ ((((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ ¬ 𝑛 = 0) ∧ ¬ 𝑛 = (𝑠 + 1)) ∧ (𝑠 + 1) < 𝑛) → 0 = ((𝑇‘(𝑏‘(𝑖 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖)))))
175		simp-4r 782	. . . . . . . . . . . . . . 15 ⊢ ((((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ ¬ 𝑛 = 0) ∧ ¬ 𝑛 = (𝑠 + 1)) ∧ ¬ (𝑠 + 1) < 𝑛) → 𝑛 = 𝑖)
176	175	fvoveq1d 7427	. . . . . . . . . . . . . 14 ⊢ ((((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ ¬ 𝑛 = 0) ∧ ¬ 𝑛 = (𝑠 + 1)) ∧ ¬ (𝑠 + 1) < 𝑛) → (𝑏‘(𝑛 − 1)) = (𝑏‘(𝑖 − 1)))
177	176	fveq2d 6892	. . . . . . . . . . . . 13 ⊢ ((((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ ¬ 𝑛 = 0) ∧ ¬ 𝑛 = (𝑠 + 1)) ∧ ¬ (𝑠 + 1) < 𝑛) → (𝑇‘(𝑏‘(𝑛 − 1))) = (𝑇‘(𝑏‘(𝑖 − 1))))
178	175	fveq2d 6892	. . . . . . . . . . . . . . 15 ⊢ ((((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ ¬ 𝑛 = 0) ∧ ¬ 𝑛 = (𝑠 + 1)) ∧ ¬ (𝑠 + 1) < 𝑛) → (𝑏‘𝑛) = (𝑏‘𝑖))
179	178	fveq2d 6892	. . . . . . . . . . . . . 14 ⊢ ((((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ ¬ 𝑛 = 0) ∧ ¬ 𝑛 = (𝑠 + 1)) ∧ ¬ (𝑠 + 1) < 𝑛) → (𝑇‘(𝑏‘𝑛)) = (𝑇‘(𝑏‘𝑖)))
180	179	oveq2d 7421	. . . . . . . . . . . . 13 ⊢ ((((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ ¬ 𝑛 = 0) ∧ ¬ 𝑛 = (𝑠 + 1)) ∧ ¬ (𝑠 + 1) < 𝑛) → ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛))) = ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖))))
181	177, 180	oveq12d 7423	. . . . . . . . . . . 12 ⊢ ((((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ ¬ 𝑛 = 0) ∧ ¬ 𝑛 = (𝑠 + 1)) ∧ ¬ (𝑠 + 1) < 𝑛) → ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛)))) = ((𝑇‘(𝑏‘(𝑖 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖)))))
182	174, 181	ifeqda 4563	. . . . . . . . . . 11 ⊢ (((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ ¬ 𝑛 = 0) ∧ ¬ 𝑛 = (𝑠 + 1)) → if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛))))) = ((𝑇‘(𝑏‘(𝑖 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖)))))
183	156, 182	ifeqda 4563	. . . . . . . . . 10 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ ¬ 𝑛 = 0) → if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛)))))) = ((𝑇‘(𝑏‘(𝑖 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖)))))
184	129, 183	ifeqda 4563	. . . . . . . . 9 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) → if(𝑛 = 0, ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛))))))) = ((𝑇‘(𝑏‘(𝑖 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖)))))
185		ovexd 7440	. . . . . . . . 9 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → ((𝑇‘(𝑏‘(𝑖 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖)))) ∈ V)
186	25, 184, 108, 185	fvmptd2 7003	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → (𝐺‘𝑖) = ((𝑇‘(𝑏‘(𝑖 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖)))))
187	186	oveq2d 7421	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)) = ((𝑖 ↑ 𝑋) · ((𝑇‘(𝑏‘(𝑖 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖))))))
188	187	mpteq2dva 5247	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖))) = (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ 𝑋) · ((𝑇‘(𝑏‘(𝑖 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖)))))))
189	188	oveq2d 7421	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (𝑌 Σg (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)))) = (𝑌 Σg (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ 𝑋) · ((𝑇‘(𝑏‘(𝑖 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖))))))))
190		nn0p1gt0 12497	. . . . . . . . . . . . . 14 ⊢ (𝑠 ∈ ℕ0 → 0 < (𝑠 + 1))
191		0red 11213	. . . . . . . . . . . . . . . 16 ⊢ (𝑠 ∈ ℕ0 → 0 ∈ ℝ)
192		ltne 11307	. . . . . . . . . . . . . . . 16 ⊢ ((0 ∈ ℝ ∧ 0 < (𝑠 + 1)) → (𝑠 + 1) ≠ 0)
193	191, 192	sylan 580	. . . . . . . . . . . . . . 15 ⊢ ((𝑠 ∈ ℕ0 ∧ 0 < (𝑠 + 1)) → (𝑠 + 1) ≠ 0)
194		neeq1 3003	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = (𝑠 + 1) → (𝑛 ≠ 0 ↔ (𝑠 + 1) ≠ 0))
195	193, 194	syl5ibrcom 246	. . . . . . . . . . . . . 14 ⊢ ((𝑠 ∈ ℕ0 ∧ 0 < (𝑠 + 1)) → (𝑛 = (𝑠 + 1) → 𝑛 ≠ 0))
196	34, 190, 195	syl2anc2 585	. . . . . . . . . . . . 13 ⊢ (𝑠 ∈ ℕ → (𝑛 = (𝑠 + 1) → 𝑛 ≠ 0))
197	196	ad2antrl 726	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (𝑛 = (𝑠 + 1) → 𝑛 ≠ 0))
198	197	imp 407	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑛 = (𝑠 + 1)) → 𝑛 ≠ 0)
199		eqneqall 2951	. . . . . . . . . . 11 ⊢ (𝑛 = 0 → (𝑛 ≠ 0 → ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))) = (𝑇‘(𝑏‘𝑠))))
200	198, 199	mpan9 507	. . . . . . . . . 10 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑛 = (𝑠 + 1)) ∧ 𝑛 = 0) → ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))) = (𝑇‘(𝑏‘𝑠)))
201		iftrue 4533	. . . . . . . . . . 11 ⊢ (𝑛 = (𝑠 + 1) → if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛)))))) = (𝑇‘(𝑏‘𝑠)))
202	201	ad2antlr 725	. . . . . . . . . 10 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑛 = (𝑠 + 1)) ∧ ¬ 𝑛 = 0) → if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛)))))) = (𝑇‘(𝑏‘𝑠)))
203	200, 202	ifeqda 4563	. . . . . . . . 9 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑛 = (𝑠 + 1)) → if(𝑛 = 0, ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛))))))) = (𝑇‘(𝑏‘𝑠)))
204	74, 35	syl 17	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (𝑠 + 1) ∈ ℕ0)
205		fvexd 6903	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (𝑇‘(𝑏‘𝑠)) ∈ V)
206	25, 203, 204, 205	fvmptd2 7003	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (𝐺‘(𝑠 + 1)) = (𝑇‘(𝑏‘𝑠)))
207	206	oveq2d 7421	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (((𝑠 + 1) ↑ 𝑋) · (𝐺‘(𝑠 + 1))) = (((𝑠 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑠))))
208	4	3ad2ant2 1134	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑅 ∈ Ring)
209		eqid 2732	. . . . . . . . . . . . . 14 ⊢ (Base‘𝑃) = (Base‘𝑃)
210	26, 7, 209	vr1cl 21732	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ Ring → 𝑋 ∈ (Base‘𝑃))
211	208, 210	syl 17	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑋 ∈ (Base‘𝑃))
212		eqid 2732	. . . . . . . . . . . . . 14 ⊢ (mulGrp‘𝑃) = (mulGrp‘𝑃)
213	212, 209	mgpbas 19987	. . . . . . . . . . . . 13 ⊢ (Base‘𝑃) = (Base‘(mulGrp‘𝑃))
214		eqid 2732	. . . . . . . . . . . . . 14 ⊢ (1r‘𝑃) = (1r‘𝑃)
215	212, 214	ringidval 20000	. . . . . . . . . . . . 13 ⊢ (1r‘𝑃) = (0g‘(mulGrp‘𝑃))
216	213, 215, 28	mulg0 18951	. . . . . . . . . . . 12 ⊢ (𝑋 ∈ (Base‘𝑃) → (0 ↑ 𝑋) = (1r‘𝑃))
217	211, 216	syl 17	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (0 ↑ 𝑋) = (1r‘𝑃))
218	7	ply1crng 21713	. . . . . . . . . . . . . . 15 ⊢ (𝑅 ∈ CRing → 𝑃 ∈ CRing)
219	218	anim2i 617	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑁 ∈ Fin ∧ 𝑃 ∈ CRing))
220	219	3adant3 1132	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝑁 ∈ Fin ∧ 𝑃 ∈ CRing))
221	8	matsca2 21913	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ Fin ∧ 𝑃 ∈ CRing) → 𝑃 = (Scalar‘𝑌))
222	220, 221	syl 17	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑃 = (Scalar‘𝑌))
223	222	fveq2d 6892	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (1r‘𝑃) = (1r‘(Scalar‘𝑌)))
224	217, 223	eqtrd 2772	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (0 ↑ 𝑋) = (1r‘(Scalar‘𝑌)))
225	224	adantr 481	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (0 ↑ 𝑋) = (1r‘(Scalar‘𝑌)))
226	225	oveq1d 7420	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → ((0 ↑ 𝑋) · (𝐺‘0)) = ((1r‘(Scalar‘𝑌)) · (𝐺‘0)))
227	7, 8	pmatlmod 22186	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑌 ∈ LMod)
228	4, 227	sylan2 593	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑌 ∈ LMod)
229	228	3adant3 1132	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑌 ∈ LMod)
230	20, 21, 7, 8, 22, 23, 2, 24, 25	chfacfisf 22347	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → 𝐺:ℕ0⟶(Base‘𝑌))
231	4, 230	syl3anl2 1413	. . . . . . . . . 10 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → 𝐺:ℕ0⟶(Base‘𝑌))
232	231, 81	ffvelcdmd 7084	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (𝐺‘0) ∈ (Base‘𝑌))
233		eqid 2732	. . . . . . . . . 10 ⊢ (Scalar‘𝑌) = (Scalar‘𝑌)
234		eqid 2732	. . . . . . . . . 10 ⊢ (1r‘(Scalar‘𝑌)) = (1r‘(Scalar‘𝑌))
235	1, 233, 27, 234	lmodvs1 20492	. . . . . . . . 9 ⊢ ((𝑌 ∈ LMod ∧ (𝐺‘0) ∈ (Base‘𝑌)) → ((1r‘(Scalar‘𝑌)) · (𝐺‘0)) = (𝐺‘0))
236	229, 232, 235	syl2an2r 683	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → ((1r‘(Scalar‘𝑌)) · (𝐺‘0)) = (𝐺‘0))
237		iftrue 4533	. . . . . . . . 9 ⊢ (𝑛 = 0 → if(𝑛 = 0, ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛))))))) = ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))))
238		ovexd 7440	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))) ∈ V)
239	25, 237, 81, 238	fvmptd3 7018	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (𝐺‘0) = ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))))
240	226, 236, 239	3eqtrd 2776	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → ((0 ↑ 𝑋) · (𝐺‘0)) = ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))))
241	207, 240	oveq12d 7423	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → ((((𝑠 + 1) ↑ 𝑋) · (𝐺‘(𝑠 + 1))) + ((0 ↑ 𝑋) · (𝐺‘0))) = ((((𝑠 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑠))) + ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0))))))
242	1, 3	cmncom 19660	. . . . . . 7 ⊢ ((𝑌 ∈ CMnd ∧ ((0 ↑ 𝑋) · (𝐺‘0)) ∈ (Base‘𝑌) ∧ (((𝑠 + 1) ↑ 𝑋) · (𝐺‘(𝑠 + 1))) ∈ (Base‘𝑌)) → (((0 ↑ 𝑋) · (𝐺‘0)) + (((𝑠 + 1) ↑ 𝑋) · (𝐺‘(𝑠 + 1)))) = ((((𝑠 + 1) ↑ 𝑋) · (𝐺‘(𝑠 + 1))) + ((0 ↑ 𝑋) · (𝐺‘0))))
243	13, 83, 96, 242	syl3anc 1371	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (((0 ↑ 𝑋) · (𝐺‘0)) + (((𝑠 + 1) ↑ 𝑋) · (𝐺‘(𝑠 + 1)))) = ((((𝑠 + 1) ↑ 𝑋) · (𝐺‘(𝑠 + 1))) + ((0 ↑ 𝑋) · (𝐺‘0))))
244		ringgrp 20054	. . . . . . . . 9 ⊢ (𝑌 ∈ Ring → 𝑌 ∈ Grp)
245	10, 244	syl 17	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑌 ∈ Grp)
246	245	adantr 481	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → 𝑌 ∈ Grp)
247	207, 96	eqeltrrd 2834	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (((𝑠 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑠))) ∈ (Base‘𝑌))
248	10	adantr 481	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → 𝑌 ∈ Ring)
249	24, 20, 21, 7, 8	mat2pmatbas 22219	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝑇‘𝑀) ∈ (Base‘𝑌))
250	4, 249	syl3an2 1164	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝑇‘𝑀) ∈ (Base‘𝑌))
251	250	adantr 481	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (𝑇‘𝑀) ∈ (Base‘𝑌))
252		simpl1 1191	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → 𝑁 ∈ Fin)
253	208	adantr 481	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → 𝑅 ∈ Ring)
254		elmapi 8839	. . . . . . . . . . . 12 ⊢ (𝑏 ∈ (𝐵 ↑m (0...𝑠)) → 𝑏:(0...𝑠)⟶𝐵)
255	254	adantl 482	. . . . . . . . . . 11 ⊢ ((𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) → 𝑏:(0...𝑠)⟶𝐵)
256	255	adantl 482	. . . . . . . . . 10 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → 𝑏:(0...𝑠)⟶𝐵)
257		0elfz 13594	. . . . . . . . . . . 12 ⊢ (𝑠 ∈ ℕ0 → 0 ∈ (0...𝑠))
258	34, 257	syl 17	. . . . . . . . . . 11 ⊢ (𝑠 ∈ ℕ → 0 ∈ (0...𝑠))
259	258	ad2antrl 726	. . . . . . . . . 10 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → 0 ∈ (0...𝑠))
260	256, 259	ffvelcdmd 7084	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (𝑏‘0) ∈ 𝐵)
261	24, 20, 21, 7, 8	mat2pmatbas 22219	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (𝑏‘0) ∈ 𝐵) → (𝑇‘(𝑏‘0)) ∈ (Base‘𝑌))
262	252, 253, 260, 261	syl3anc 1371	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (𝑇‘(𝑏‘0)) ∈ (Base‘𝑌))
263	1, 22	ringcl 20066	. . . . . . . 8 ⊢ ((𝑌 ∈ Ring ∧ (𝑇‘𝑀) ∈ (Base‘𝑌) ∧ (𝑇‘(𝑏‘0)) ∈ (Base‘𝑌)) → ((𝑇‘𝑀) × (𝑇‘(𝑏‘0))) ∈ (Base‘𝑌))
264	248, 251, 262, 263	syl3anc 1371	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → ((𝑇‘𝑀) × (𝑇‘(𝑏‘0))) ∈ (Base‘𝑌))
265	1, 2, 23, 3	grpsubadd0sub 18906	. . . . . . 7 ⊢ ((𝑌 ∈ Grp ∧ (((𝑠 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑠))) ∈ (Base‘𝑌) ∧ ((𝑇‘𝑀) × (𝑇‘(𝑏‘0))) ∈ (Base‘𝑌)) → ((((𝑠 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑠))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))) = ((((𝑠 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑠))) + ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0))))))
266	246, 247, 264, 265	syl3anc 1371	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → ((((𝑠 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑠))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))) = ((((𝑠 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑠))) + ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0))))))
267	241, 243, 266	3eqtr4d 2782	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (((0 ↑ 𝑋) · (𝐺‘0)) + (((𝑠 + 1) ↑ 𝑋) · (𝐺‘(𝑠 + 1)))) = ((((𝑠 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑠))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))))
268	189, 267	oveq12d 7423	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → ((𝑌 Σg (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)))) + (((0 ↑ 𝑋) · (𝐺‘0)) + (((𝑠 + 1) ↑ 𝑋) · (𝐺‘(𝑠 + 1))))) = ((𝑌 Σg (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ 𝑋) · ((𝑇‘(𝑏‘(𝑖 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖))))))) + ((((𝑠 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑠))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0))))))
269	113, 268	eqtrd 2772	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (((𝑌 Σg (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)))) + ((0 ↑ 𝑋) · (𝐺‘0))) + (((𝑠 + 1) ↑ 𝑋) · (𝐺‘(𝑠 + 1)))) = ((𝑌 Σg (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ 𝑋) · ((𝑇‘(𝑏‘(𝑖 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖))))))) + ((((𝑠 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑠))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0))))))
270	75, 102, 269	3eqtrd 2776	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (𝑌 Σg (𝑖 ∈ (0...(𝑠 + 1)) ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)))) = ((𝑌 Σg (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ 𝑋) · ((𝑇‘(𝑏‘(𝑖 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖))))))) + ((((𝑠 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑠))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0))))))
271	40, 73, 270	3eqtrd 2776	1 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (𝑌 Σg (𝑖 ∈ ℕ0 ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)))) = ((𝑌 Σg (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ 𝑋) · ((𝑇‘(𝑏‘(𝑖 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖))))))) + ((((𝑠 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑠))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0))))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   ∨ wo 845   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   ≠ wne 2940  Vcvv 3474   ∪ cun 3945   ∩ cin 3946  ∅c0 4321  ifcif 4527  {csn 4627   class class class wbr 5147   ↦ cmpt 5230  ⟶wf 6536  ‘cfv 6540  (class class class)co 7405   ↑_m cmap 8816  Fincfn 8935  ℂcc 11104  ℝcr 11105  0cc0 11106  1c1 11107   + caddc 11109   < clt 11244   ≤ cle 11245   − cmin 11440  ℕcn 12208  2c2 12263  ℕ₀cn0 12468  ℤcz 12554  ℤ_≥cuz 12818  ...cfz 13480  Basecbs 17140  +_gcplusg 17193  ._rcmulr 17194  Scalarcsca 17196   ·_𝑠 cvsca 17197  0_gc0g 17381   Σ_g cgsu 17382  Mndcmnd 18621  Grpcgrp 18815  -_gcsg 18817  ._gcmg 18944  CMndccmn 19642  mulGrpcmgp 19981  1_rcur 19998  Ringcrg 20049  CRingccrg 20050  LModclmod 20463  var₁cv1 21691  Poly₁cpl1 21692   Mat cmat 21898   matToPolyMat cmat2pmat 22197

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-tp 4632  df-op 4634  df-ot 4636  df-uni 4908  df-int 4950  df-iun 4998  df-iin 4999  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-se 5631  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-isom 6549  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-of 7666  df-ofr 7667  df-om 7852  df-1st 7971  df-2nd 7972  df-supp 8143  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-1o 8462  df-er 8699  df-map 8818  df-pm 8819  df-ixp 8888  df-en 8936  df-dom 8937  df-sdom 8938  df-fin 8939  df-fsupp 9358  df-sup 9433  df-oi 9501  df-card 9930  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-nn 12209  df-2 12271  df-3 12272  df-4 12273  df-5 12274  df-6 12275  df-7 12276  df-8 12277  df-9 12278  df-n0 12469  df-z 12555  df-dec 12674  df-uz 12819  df-rp 12971  df-fz 13481  df-fzo 13624  df-seq 13963  df-hash 14287  df-struct 17076  df-sets 17093  df-slot 17111  df-ndx 17123  df-base 17141  df-ress 17170  df-plusg 17206  df-mulr 17207  df-sca 17209  df-vsca 17210  df-ip 17211  df-tset 17212  df-ple 17213  df-ds 17215  df-hom 17217  df-cco 17218  df-0g 17383  df-gsum 17384  df-prds 17389  df-pws 17391  df-mre 17526  df-mrc 17527  df-acs 17529  df-mgm 18557  df-sgrp 18606  df-mnd 18622  df-mhm 18667  df-submnd 18668  df-grp 18818  df-minusg 18819  df-sbg 18820  df-mulg 18945  df-subg 18997  df-ghm 19084  df-cntz 19175  df-cmn 19644  df-abl 19645  df-mgp 19982  df-ur 19999  df-ring 20051  df-cring 20052  df-subrg 20353  df-lmod 20465  df-lss 20535  df-sra 20777  df-rgmod 20778  df-dsmm 21278  df-frlm 21293  df-ascl 21401  df-psr 21453  df-mvr 21454  df-mpl 21455  df-opsr 21457  df-psr1 21695  df-vr1 21696  df-ply1 21697  df-mamu 21877  df-mat 21899  df-mat2pmat 22200

This theorem is referenced by:  cpmadugsum  22371